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106        XIII    INTEGRATION

elsewhere (/= 1, 2). Then /?f is continuous on X, because /f/(/i +/2) is
continuous at the points x where f^x) Jrf2(x) > 0, and also |/Zj(x)| < |/?(x)|
for all jc G X, which proves that ht is continuous at the points x where
/iOO +/2(X)= 0, because /?(*) also vanishes at these points. It is clear
that | A, | <*/i (z = 1, 2) and that h = hi + h2; hence

Since |ju(/?)| may be taken to be arbitrarily close to pC/i + /2), it follows that
P(/i + /2)  P(/I) + p(/2). Hence ( is proved.

We shall now extend the definition of />(/) to all functions/e JfR(X). To
do this we write p(/) = p(f) - p(/")> where/=/'  f" is any decomposition
of /as the difference of two functions/',/" ^ 0 belonging to JfR(X). The
value of p(/) so obtained is independent of the decomposition, because if
/=/i -/? =/2 -/!'> then // +/J =/^' +/2' and therefore p(f[) + p(/^) =
P(/D + P(/2> by virtue of (13.3.12).

With this definition, the formula ( is valid for all/j.,/2 in JfR(X).
For we can write/! =// -f'^f2 =f2 -f where// 9f? (i = 1, 2) are ^0 and
belong to Jf R(X); since /x +/2 = (// +/2f) - t/i" +/20 our assertion follows
from the definition above and from ( for functions ^0.

Finally, the above definition shows that, for each scalar a ^ 0, we have
p(af) = ap(f); and if a < 0, then we have

p(af) = p(af - of) = p(-aD + p(af)

Hence the relation p(tz/) = ap(f) is valid for all real scalars a, and therefore
we have proved that p is a (real) linear form on JTR(X). Hence by (13.3.1) it
is a positive measure.                                                                Q.E.D.

The measure p defined by (13.3.2) is called the absolute value of the
complex measure //, and is denoted by |/*|. Hence by definition we have

(133.3)                |M/)|  l/ild/l)       for all   /e Jfc(X).

It is immediately seen that if a e C and p is a measure on X, we have

(13.3.4)                                   M = H'M.
If i*. is a positive measure on X, then

(13.3.5)                                     |,i| = /i.measure on Ua such that, for each pair of indices a, j8, the restrictions of i*a